Department of Mathematics
UNIVERSITY OF ROCHESTER
Tel (716) 275-4429 or 244-9368
FAX (716) 244 6631 (at U of R)
Ralph A. Raimi
To the Editors,
Journal for Research in Mathematics Education:
1906 Association Drive
Reston, VA 20191-9988
Mathematicians Writing, by Leone Burton and Candia Morgan, in volume 31 of JRME (July, 2000) pp 429-453, contains some errors that should be noted in JRME, and amended before their interesting research is pushed much further. I refer in particular to the Indicators of Authority that came under discussion on page 439-440.
"One important aspect of the identities of authors as projected by their texts is the extent to which and the manner in which they claim to be authoritative within their community.....
"Terms such as clearly and obvious … [serve] as a claim to authority on the part of the writer (implying that this derivation is clear to me and I do not need to explain it further because if it is not clear to you, that is your fault not mine) ...... We believe that, whatever the author's intent, the extent or absence of such words is one of the interpersonal aspects of the writing that will influence the ways in which the readers of the text will construct an image of the author and will consequently judge the worth of the text itself."
Now it might be that -- "whatever the author's intent" might be -- some readers might judge the character of the author and the worth of his work under the influence of having seen terms such as obvious in that work; but to say -- as Burton and Morgan clearly do in the immediately preceding sentences -- that using such a term constitutes a claim to authority is not correct.
It is by authority that I know that Napoleon was defeated at Waterloo in 1815, as I have no firsthand knowledge of this alleged fact, only the statements (albeit rich and interlocking) of persons I believe. We all make use of authority in this way every minute of our lives, and give it little thought. But in the writing of mathematical papers, there is never an appeal to authority.
Mathematical statements are almost unique in their failure to appeal to authority. If there is a gap in an argument, the word obvious does the very opposite of concealing the gap, and is not asking the reader to accept the writer's word on the matter. Obvious and clearly merely abbreviate the discussion for the benefit of the persons who already know the connecting link that is being omitted.
Of course it is a matter of judgment for the author to decide what parts of his argument may be omitted for expository purposes, and some authors presuppose more skill or previous knowledge on the part of the reader than others do. Sometimes the assessment of audience fails; it has often been an occasion of merriment among mathematicians when someone (in print or at the blackboard) imagines obvious that which is not -- not even, on reflection, to the author himself. But to imagine that the author is expecting his reputation to overwhelm the reader by using such phrases for intimidation is simply mistaken. There is no form of literature less "authoritative" than mathematics.
On the following page, a few sentences later, Burton and Morgan write,
"Indicators of authority claims from the use of modifiers, such as clearly, easily, of course, and immediately obvious, and phrases, such as without loss of generality, it suffices to consider the case, and the last stage is trivial, all signal a gap in the argument (implying that there is no accompanying justification for the claim that generality is not lost) and hence signal to the readers that they should accept the author's claim."
There are many things that can be said about the use of such phrases to abbreviate a mathematical discussion, but that they "signal to the readers that they should accept the author's claim" on the author's authority is not one of the correct ones.
"Without loss of generality", for example, is a technical phrase used in consideration for the reader (and publisher). To consider it an assertion of authority rather than a part of the proof is analogous to considering "Let x be an element of G" an intrusion on the reader's liberty to consider x in other ways; no mathematician would imagine that the apparently peremptory tone of "Let x be an element of G" constitutes a claim to authority.
An explanation (by example) of without loss of generality might be of value here, for it would surely not be wise to ask the present reader to accept my preceding paragraph on my own authority, either: In proving by use of a coordinate system the theorem that the medians of a triangle meet at a point, a mathematician might begin by saying, "Let ABC be the triangle. Without loss of generality, suppose A is at the origin (0,0) and AB is along the x-axis with B at the point (b,0)."
It is thus asserted, in other words, that proving the theorem for this specially placed triangle proves it for all other triangles. Had the proof begun by asking, in addition, that the point C be given the coordinates (b,b), the mathematician could be called to order for inappropriate use of "WOLOG" (a common informal abbreviation for without loss of generality, so frequent and so little idiosyncratic is the use of the term), for the triangle thus specified would a right triangle, and proving things for a right triangle is not proving it for all triangles.
Why, on the contrary, does placing two of the vertices at (0,0) and (0,b) fail to lose generality? Surely not every triangle in the plane occupies so particular a position. Isn't this mathematician cheating a bit, making it easier for himself by this ploy, intimidating his less secure readers by some claim to authority?
Well, it is the genius of the entire method of coordinates, that the Euclidean properties of Euclidean figures remain invariant under rigid motions, that concurrence and distances are Euclidean properties in this sense, and that there is always a rigid motion which will bring any triangle, wherever placed initially, to the position described in this proof.
Now the whole story of how Euclidean geometry can be got at via coordinate systems is really quite long Not everyone has learned these things, and a high school student using analytic geometry for the first time often does take such missing links on faith, sometimes not even noticing that anything is missing. If the mathematician were writing for that high school student he might well avoid "WOLOG" and provide a longer explanation. Yet mathematical and scientific exposition would come to a stop if all papers began "at the beginning." Mathematical research papers, especially, are father from first principles than most expositions because directed at professional mathematicians.
On the other hand, the path back, from where the expositor is now, to where the first principles are to be found, must always be visible. In tackling this theorem about the centroid of a triangle my imagined author is presuming the reader has already come a certain distance in analytic geometry. Yet rather than merely placing the triangle's base on the x-axis with a vertex at the origin, this particular exposition makes a point of reminding the reader, via "WOLOG", that there is no sleight-of-hand at work.
Far from being an appeal to authority, the phrase without loss of generality is a reminder that some "first principles," or some of the intervening logical paths, are being omitted, so that the uncomprehending reader, who might otherwise think himself obtuse for not seeing why the proof for the special case proves them all, is given the more pleasant information that he is merely missing some information. At the same time, the comprehending reader, the author's expected audience, is saved some tedium. So with the other phrases cited by Burton and Morgan as assertions of authority. Take it from me, they are no such thing.
Ralph A. Raimi